Reference Book on Special Functions
The Bateman Manuscript Project is a good source, as is the NIST Digital Library of Mathematical Functions.
As OP stated he was a physics student in a comment, there are a lot of books called (something similar) to "Mathematical Methods for Physicists", which have relevant references, details and techniques for physicists. I'd probably start with the first two.
Arfken, "Mathematical Methods for Physicists"
Boas, "Mathematical Methods in the Physical Sciences "
Courant and Hilbert, "Methods of Mathematical Physics"
Shankar, "Basic Training in Mathematics: A Fitness Program for Science Students" (probably too basic)
Abramowitz and Stegun, Handbook of Mathematical Functions (not proofs, but there are a lot of useful things there, which you can look up elsewhere)
(Website) Wolfram Mathworld
And for PDE's, you can look at specific PDE books:
Zauderer, Partial Differential Equations of Applied Mathematics
One of the many basic linear PDE books, e.g. Haberman's Applied PDE
You may consider looking at Oldham, Myland, and Spanier's book An Atlas of Functions which can be purchased used quite cheaply on Amazon. I would dare suggest it is a beautiful book, with very nice graphs on glossy pages. It is large and it is organized by classes of functions. There aren't many proofs in the book, so I'm not sure if that would deter you from this book.
Another route to go is to get Schaum's Outline of Mathematical Formulas, though it may not be detailed enough for you.
One resource I haven't seen mentioned is the trusty Table of Integrals, Series, and Products by Gradshteyn & Ryhzik. While known primarily as an integral table, this book has a ton of other material including good coverage of special functions.
Further, I second (or third, etc) the suggestion of the Handbook of Mathematical Functions by Abramowitz & Stegun. It is amusing to read a book with numerical tables, however.
A book I really appreciated is
Lebedev - Special function and their applications
Dove edition.